CHAPTER 114 EXCITATION OF WAVES INSIDE A BOTTOMLESS HARBOR

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1 CHAPTER 114 EXCITATION OF WAVES INSIDE A BOTTOMLESS HARBOR by Nbru Sakua, Jhannes Bu'hler and R. L. Wiegel INTRODUCTION In planning enclsed areas in the cean, such as ffshre harbrs fr fishing r recreatinal bats, ne has t cnsider very carefully the prble f frced seiches due t surface water waves. A nuber f papers have been written n this prble> but t the authrs' knwledge, nly ne f the has treated the case f an artificial bttless circular harbr in the pen cean, fr which the walls extend nly part way t the sea flr. A btt t such a harbr, which wuld be expensive, can nly be itted if the effect f its absence n the sea surface inside the harbr is nt verly detriental. C.J.R. Garrett (1970) ade a theretical study f the excitatin f waves inside such a harbr fr the 'n-entrance' case. The bject f this caper is t evaluate experientally the behavir f the sea surface inside such a harbr, t cpare the results with the theretical values btained by Garrett, and t extend the bservatins t the case fr which there is an entrance t the harbr. 2. THEORY A thery f the excitatin f waves inside a partially iersed bttless vertical circular cylinder was develped by C.J.R. Garrett. In this thery, the free surface displaceent ay be described by real partf Q ~ lcrt J where the incident wave is given by C = C e lkx (2.1) = C / e i J (kr) cs 0 (2.2) L- ^Kj where e = 1, and e - 2 fr 1; is the aplitude f the incident wave. Disturbances within a circular cylinder can be expressed apprxiately by Research Engineer, Public Wrks Research Institute, Ministry f Cnstructin, Japan (frerly graduate student, Dept. f Civil Engrg., Univ. f Califrnia, Berkeley, Califrnia Graduate student, Dept. f Civil Engrg., Univ. f Califrnia, Berkeley, Califrnia Prfessr, Dept. f Civil Engrg., Univ. f Califrnia, Berkeley, Califrnia 2005

2 2006 COASTAL ENGINEERING C (r,6) = 0 2. e 1 %n (r) cs e 2-3 > The crrespnding displaceent ptential (the prduct f and the velcity ptential) is 1CT (r,0,) = J C V e i \ji 4 (r,) cs e (2.4) ) ' where V r) =ar U 2-5 > In rder t satisfy the kineatic free surface cnditin, ep ust als satisfy 2 7 cp = 0 (2.6) O tp - g ^ = 0 n = d (2.7) - -^ = 0 n = 0 (2.8) -5-2 = 0 n r = a fr h d (2.9) 3 r Garrett fund that the apprpriate expansin f ty culd be expressed as fllws. In r ^ a J'(ka) Z fc () t i T (r,) = J (kr) -,' H (kr) -rjs H (ka) Z () K (r) K' (a) 5_ F :.,..,.. ff (2.10) J, Y ; rdinary Bessel functins I, K ; dified Bessel functins H = J + iy ; the Hankel functin f the first kind -1 Z, () = N 8 csh k (2.11) k k _l Z () = N " cs ^ (2.12)

3 BOTTOMLESS HARBOR 2007 N, = i [1 + (sinh 2kd/2kd)] (2.13) k N = \ [1 + (sin 2d/2ffd )] (2.14) a h F = - f f () Z () d (2.15) a i t> a In r r i_(ar) ill (r,) = ) F TT7 r Z () Y Z_ cv I (aa) a (2.16) Using Eqs. (2.5) and (2.16), the free surface elevatin f Eq. (2.3) is, fr r a ' I («r) V (r) = A J (kr) + ) F -pr? r '(d), n. Si Z. al (aa) «(2.17) F Z^(d> k kj' (ka) (2.18) The slutin fr r s a ay be written where v_(r) = J (kr) + B H (kr) + S F \^ '(d) (2.19) Ti /_. ff T,,., y Q, u K (aa) j'(ka) B = (A - 1) T^-TT- T (2.20) H (ka) The first ter n the right hand side f Eq. (2.17) is the st iprtant cntributin t the wave tin inside the cylinder; the ther ters describe waves which are generally cnfined near r - a. Garrett ade nuerical calculatins f A fr several cnditins. These have been reprduced in Figs. 1 and 2, rder that they can be cpared with experiental results. 3. EXPERIMENTAL ARRANGEMENTS AND PROCEDURE The experiental study reprted herein was ade in the hydraulic del basin (150 ft. by 63 ft. by 2j ft.) at the Richnd Field Statin f the University f Califrnia. 3.1 Cnditins Tested The cnditins that were tested in the experient are as fllws (fr definitin f the sybls, see Fig. 3):

4 2008 COASTAL ENGINEERING * ""^ *\ 1 " / 1 1 / It sit > CD Si d II d II LU a: t- LU => O X 1- S e \ s _/ \'/ \ \ \ c ii n i i i i i Ii ' 4 > t\l "N ^^ ' \ ' y : LU ii s LU ID 1- g II X g N s/ [7 E d

5 BOTTOMLESS HARBOR 2009 CASE NO. d a h a NOTES Withut entrance With an entrance in the rear With an entrance n the side With an entrance at the frnt. with d 0.9 ft.; 1.5 f t. ; h =0.3 f t., and ka = 2 /L = Tw different experiental arrangeents were used. The first series f experients were cnducted in a sectin f the del basin fr cases 1 and 2. This arrangeent is shwn in Fig. 4 (sall basin). The experients were then repeated with an arrangeent as shwn in Fig. 5 (large basin) and the results cpared. Because f the reduced reflectin effects, the reaining cases were then studied in the large basin. 3.2 Experiental Arrangeents and Prcedures Wave heights and wave perids inside and utside the cylinder were easured by eans f parallel wire resistance type wave gages and a ultichannel rectilinear writing scillgraph (Wiegel, 1953). Fur wave gages were used t easure the waves; ne was used t easure the incident waves, and three were used t easure the waves inside the cylinder. The wave gage that was used fr easuring incident waves was installed as far away fr the cylinder as pssible in the del basin in rder t iniie the effect f wave reflectin fr the cylinder (see Figs. 4 and 5). The ther three wave gages were installed inside the cylinder t enable easureent f the axiu wave height fr each de f harbr water surface scillatin. In easuring the frced surface scillatins inside the cylinder and in interpreting the results, it is useful t cnsider the des f scillatin t be the sae as the des f free scillatin inside a vertical circular harbr which extends t the btt. This case has been studied theretically and experientally by J. S. McNwn (1951; 1952) and by Gda (1963). The water surface elevatins inside a vertical circular cylinder are expressed as C B = C J (kr) cs (3.1)

6 2010 COASTAL ENGINEERING 5 GO r Id O a te X C^//, ////,{/ {//X- {-(*.. ^ '' ww f UQ l-u (TV) ujk > p at Di t; i/i r ^:

7 BOTTOMLESS HARBOR 2011 where C is a cefficient related t the wave aplitude, k is the wave nuber (2TT/L) > L is the incident wave length, r is the radial crdinate, 6 is the angular crdinate, is an integer and J is the Bessel's functin f rder. Se f the pssible des expressed by the abve expressin are shwn in Fig. 6, In the experient, hwever, it appeared that the de r des ccurring within the cylinder were stly f the lwest rder f ; in ther wrds, 0-» 1, 2, 3 (Fig. 10) were bserved, even thugh theretically the nuber f des pssible fr a given value f ka is infinite. The places where the axiu wave elevatin were bserved in the cylinder were either at the center f the cylinder r near the wall f the cylinder. The rientatin f the ndal lines f de 1, fr instance, wuld be nral t the directin f advance f incident waves. Therefre, fr alst all f the experiental runs the three gages inside the cylinder were installed as shwn in Fig. 7. The cylinder which was used as a del f a circular harbr was ade f steel and had three legs t hld it at the desired elevatin abve the btt. After a nuber f easureents were ade, an entrance t the harbr was ade by cutting ut a sectin f the cylinder, as shwn in Fig. 8. Fr additinal easureents withut an entrance, the sectin was put back in place and the gaps clsed with a plastic sealant. An exaple f an incident wave recrd is shwn in Fig. 9. It was difficult t define a "height f the incident wave," especially at higher wave nubers ka. When the generatr was started, the waves gradually increased in height. The effects f reflectin becae evident befre a unifr height was reached. The incident wave height was defined as the equilibriu wave height after a lng tie. This includes reflectins fr the walls f the basin and the cylinder itself. The final tests were ade in the large basin in rder t iniie the effect f these reflectins. Reprductins f the recrds f representative saples f nearly all cnditins have been reprduced in a labratry reprt (Sakua, Buhler and Wiegel, 1971). Fr this incident wave height, the crrespnding values f AjJ can be calculated as fllws. Fr Eq. 2.17, except near the wall, X (r) can be apprxiated by X (r) = A J (kr) (4.1) as the first ter n the right hand side f Eq. 2.17, A J (kr), is a uch re iprtant cntributin t the wave tin inside the cylinder than is the secnd ter. Substituting Eq. 4.1 int Eq. 2.3 results in CO ^ (r,6) 8 ) e i A J (kr) cs 6 (4.2) L.

8 2012 COASTAL ENGINEERING U u > i s a: in Ul-J LU C3

9 BOTTOMLESS HARBOR ul M-.._;!_a?[/.:... c E r cr u. > _l > _1 CD I- cr 3 i 4..j. t.^.r~-h u :-.r^nf?. u> i~: : ~~. h^~ O i_tjj3~ '';- ^rfe~l S Q O Z in > 5 d> t CO CVl CL 10 O _] II > I- Q u. _) 0. X en Mi:0

10 2014 COASTAL ENGINEERING C(r.e) C v ) e i J (kr) 'cs 8 L* ^ (4.3) where g = 1, and e = 2 fr s 1. Defining A' as s (4.4) Z_. ax =^> with [cs 6] - 1, and t cylinder. (r, 0) is the axiu wave height in the When =, Eq. 4.4 will be C (r,6) / _ TII ax * ax (4.5) Tnax (4.6) since [J (kr) ax = 1 When = 1: C (r, 9) A'= ^L (4.7) 4 = 2 [^(kr) ] C (r,6) _ Tnax ~ 2(0.58) Q (4.8) C (r, 6) = 0.86 * (4.9) when = 2: C Tnax (r,9) *2 2 C CJ (kr) ] *b 2 ax (4.10) C (r,9) Tnax 2 [J (kr) 3 ax (4.11) (r,8) (4.12)

11 BOTTOMLESS HARBOR 2015 when = 3 ^ (rb) A 3 = 2 C "[l( k r) 4-13 > B 3 ax C" (r,6) 2(0.44) C 4.14) i (4.15) and s n. As stated previusly, the des f scillatin which were bserved in the cylinder were stly f the J, J.., J, J_ types, r a cbinatin f tw r re f these des. The fact that higher des f scillatin were nt bserved, cbined with the likely accuracy f the experiental easureents, A^ can be apprxiated by: %; (r,9) ax (4.16) fr st practical purpses. This apprxiatin was used t calculate A^ as shwn in Figs RESULTS AND DISCUSSION 4.1 Sall Basin The results btained fr the sall basin are shwn in Figs. 12 and 13. Each bserved de f scillatin is assigned a sybl. The sybls are defined in Fig. 10, The des were recgnied by a cbinatin f visual bservatin and a study f the recrdings f the utputs f the three wave gages within the cylinder. Generally, it was very difficult t tell which de r des f scillatin were ccurring in the cylinder; seties tw r re were bserved at the sae tie, especially fr the case f the cylinder with an entrance. As stated befre, the des recgnied in the cylinder were either ne f the lwest rder pssible r a cbinatin f tw r re such des. Mdes f higher rders ight have existed in the cylinder in the experient but they were nt clear enugh t be recgnied. The rientatin f the bserved ndal lines were as shwn in Fig. 10. One has t be careful abut ^ax. Fr de IQ, fr exaple, ne wuld expect that the wave height at the lcatin where wave gage #2 was installed wuld be the sae as the wave height at wave gage #4 fr the n entrance" case. Hwever, it was fund that the wave height at wave gage #4 was larger than the height at wave gage #2 except fr ka >4.5 with h/a = 0.2 (Fig. 11, Sall Basin). It is iprtant t nte in this regard that McNwn (1951; 1952) fund such a nn-syetry fr the case f waves with a perid clse t tw resnant cnditins having nearly equal perids.

12 2016 COASTAL ENGINEERING D t- -r- tr. HI ci d ii ii j= -c\ 9 «** J «« i? (j 1 n Q Z > 5. tn i _j _j Ul T 0 3 ^0 0 J r 0 O CO l > *-n wi O CD 10 - ci d Q >

13 BOTTOMLESS HARBOR 2017 In calculating values f A^, the axiu value f Ef bserved at any ne f the 3 gages was used. Cnsider the theretical and experiental results fr case 1 with h/a ~ 0.2 as shwn in Figs. 1 and 12. The resnance patterns are alike; the values f ka fr which peaks fr different des f scillatin appear in Fig. 12 are very siilar t thse in Fig. 1. The calculated respnse curves shw very sharp peaks fr the larger values f ka, while a substantial daping effect is apparent in the experient data. In regard t the results fr case 2 with h/a =0.4 (see Figs. 2 and 13), the daping effect is saller than fr the case f h/a = 0.2. In additin, the rati f the wave height at wave gage #4 t the wave height at wave gage #2 is larger cpared with the case f h/a =0.2 (Fig. 11). The axiu value f A is abut the sae fr bth cases. A^ = 2.5 ~ Large Basin The results fr cases 1 and 2 are shwn in Figures 12 and 13. The des f scillatin agreed well with the nes bserved in the sall basin fr crrespnding wave nubers. The peak values f A' appear at the sae values f ka as fr the sall basin. The values f A' fr ka > 3 are saller fr the large basin. This is ainly due t reflectins fr the back wall f the sall basin. As third and furth case, a bttless, vertical circular cylinder with an entrance in the rear was studied (see McNwn, 1951 and 1952, fr a theretical and labratry study f a circular cylindrical harbr extending t the btt). The authrs are nt aware f any theretical studies f this case. Figs. 14 and 15 shw the experiental results, with n designatin fr different des f scillatin as the des bserved in the cylinder were nt clear. The surface scillatins within the cylinder appeared t be a cbinatin f tw r re des, and, in additin, were affected by the disturbances due t the flw thrugh the entrance. It is interesting t nte that a ndal line did nt fr at the entrance; instead, quite a large aplitude f scillatin was bserved. Wave diffractin was bserved at the entrance. The wave heights are generally saller than fr the harbr withut entrance except at high wave nubers fr h/a = 0.2 (Fig. 14). The peak value at ha =4 in Fig. 14 is nt well defined, aplitude f the incident wave was fluctuating. Figures 16 and 17 shw the behavir f the harbr with the entrance in different lcatins. Fr the entrance in the rear, the scillatins inside the harbr decrease rapidly with increasing wave nuber. Fr the entrance at the frnt, the scillatins are larger at high wave nubers. The high peaks near ka = 4 and ka = 7 are nt well defined. The axiu and iniu values f 5 runs are cnnected by vertical bars t give an estiate f the width f the fluctuatins. The recrdings f bth incident

14 2018 COASTAL ENGINEERING

15 BOTTOMLESS HARBOR 2019 ^ cc. tr ^

16 2020 COASTAL ENGINEERING wave height and harbr scillatins shwed large aplitude fluctuatins at these values f ka. It was evident fr bservatins ade during the tests that reflectins were the cause f this behavir. Fig. 18 shws the axiu values f A fr 3 different gage arrangeents fr the harbr with the entrance n the side. The largest scillatins f the water level ccur in the center f the harbr fr lw wave nubers and at the rete end f the harbr fr high nes. It is thus cncluded that even fr the case f an entrance n the side, the arrangeent f the gages parallel t the directin f wave incident gives the axiu wave heights. 5. CONCLUSIONS 5.1 Harbr Withut Entrance 1) Waves having the values f ka fr which the resnance peaks f the J» J-. j J, and J des ccurred, were generated in the labratry, as well as waves f a series f ther values f ka. Resnant peaks were bserved in the labratry study fr the sae values f ka at which the theretical resnant peaks ccur. 2) Daping f the wave aplitudes within the cylinder was fund t be relatively large in the labratry experients fr the larger values f ka. 3) The results f the experients ade with the cylinder withut an entrance shw the axiu values f A' t be fr 2.0 t 2.5 (large basin). The axiu values ccur at wave nubers ka saller than Harbr With an Entrance 1) The des f scillatin were usually cplex and culd nt be classified accrding t Fig ) The easureents shwed that the scillatins were saller fr an entrance in the rear than fr all entrance at the frnt, especially fr wave nubers ka larger than ) With an entrance in the rear, the axiu values f A were 1.1 fr h/a and 1.6 fr h/a ~ 0.2. The axiu values were bserved at wave nubers ka saller than ) With an entrance n the side, a axiu value f A was bserved at a wave nuber ka between 2.0 and 3.0. This axiu value f a' was 2.0 fr h/a = 0.4 and 3.2 fr h/a = ) With an entrance at the frnt, large aplitude fluctuatins were bserved near ka =4.0 and ka = 7.0. As an exaple, the values f a' ranged fr 2.4 t 5.3 at ka = 6.8 fr h/a = 0.4. The fluctuatins were due t the reflectin f waves fr the bundaries f the basin.

17 BOTTOMLESS HARBOR RESPONSE WITH 3DIFFERENT GAGE ARRANGEMENTS ARRANGEMENT; 4> SYMBOL; NUMBERS INDICATE GAGE AT WHICH MAX. HEIGHT IS MEASURED JT_ a «0.4 d a = ka FIG.I8 EXPERIMENTAL VALUES OF A vs ka

18 2022 COASTAL ENGINEERING 6. ACKNOWLEDGEMENTS The senir authr gratefully wishes t acknwledge the supprt f the Japan Public Wrks Research Institute, which ade it pssible fr hi t undertake graduate studies at the University f Califrnia. The wrk reprted herein was partially supprted by the Castal Engineering Research Center, Crps f Engineers, U.S. Ary. 7. SYMBOLS a = radius f circular cylinder A = defined by Eq. (2.18) B = defined by Eq. (2.20) C = a cefficient related t the wave aplitude in Eq. (3.1) d = water depth g - acceleratin f gravity h = vertical distance fr sea flr t btt f cylinder H = Hankel functin f the first kind I = dified Bessel functin J = rdinary Bessel functin k = wave nuber, 2TT/L L ~ wave, length = an integer, 0,1,2... r - radial crdinate T = wave perid x - hrintal crdinate in the directin f incident wave advance - vertical crdinate, = 0 at undisturbed water surface a 2 0/ -a real psitive slutin f > tan d + ~ 0 g e = a nuber: g =1. and e -2 fr > 1 = water surface elevatin = aplitude f incident waves 0 = angular crdinate

19 BOTTOMLESS HARBOR 2023 ) = suatin ver a including y + -ik y t v ; = suatin ver nly the real rts, i.e., # - - ik is excluded a. a = wave frequency, 2TT/T tp(r,0,) = displaceent ptential X(r) - defined by Eqs. (2.17) and (2.19) f (r,) - defined by Eq. (2.10) 8. REFERENCES Garrett, C.J.R., "Bttless Harbrs," Jurnal f Fluid Mechanics, Vl. 43, Part 3, pp , Gda, Y., "On the Oscillatins in a Rectangular Harbr and a Sectr- Shaped One," The 10th Castal Engineering Cnference, Japan, 1963, in Japanese. McNwn, J.S., Sur l'entretien des Oscillatins des Eaux Prtuaires, sus 1'Actin de la Haute Mer., These, Faculte des Sciences, Univ. Grenble, France, August McNwn, J.S., "Waves and Seiche in Idealied Prts," Gravity Waves, U.S. Dept. f Cerce, Natinal Bureau f Standards, Circular N. 521, 1952, pp Sakua, Nbru, Jhannes Buhler and R. L. Wiegel, Experiental Study f the Excitatin f Waves Inside a Bttless Harbr, Technical Reprt HEL 1-17, Hydraulic Engineering Labratry, University f Califrnia, Berkeley, Califrnia, May 1971, 36 pp. Wiegel, Rbert L., Parallel Wire Resistance Wave Meter, Castal Engineering Instruents, Cuncil n Wave Research, The Engineering Fundatin, 1953, pp Wiegel, Rbert L., Oceangraphical Engineering, Prentice-Hall, Inc., Englewd Cliffs, N.J., ~

Physics 321 Solutions for Final Exam

Physics 321 Solutions for Final Exam Page f 8 Physics 3 Slutins fr inal Exa ) A sall blb f clay with ass is drpped fr a height h abve a thin rd f length L and ass M which can pivt frictinlessly abut its center. The initial situatin is shwn